Lehrer used it in his act (and recorded it). The Kingston Trio heard it performed by Lehrer at The Hungry I, incorporated it into their own act, recorded it, and had a big hit with it. But Sheldon Harnick—who would later write the lyrics for Fiddler on the Roof, The Apple Tree, She Loves Me!, The Rothschilds, and others wrote it.
I guess it's along the same lines as "Everything is Broken", but kind of a different take.